If your carrots are going soft after just a few days, you're not storing them properly. Mine keep for weeks and weeks. I leave them in the plastic bag, and keep that in one of the vegetable drawers in my fridge. How are you storing yours?
A rubbery soft carrot isn't bad for you, it just isn't very pleasant. A slimy carrot is bad for you, don't eat it.
While not directly addressing the question of apple sauce, this Food Lab article by Kenji Alt on apple pie has some excellent information on differences of apple varieties. He surveys a number of common (at least in the US) varieties.
He notes that apples which brown more quickly tend to be both less tart (as acid, which underlies a tart flavor) inhibits browning; they also tend to cook down less quickly as the acid helps maintain the pectin structure.
Many apple sauce recipes try to find a balance among several apples, with both tart and sweeter varieties, and some that break down quickly for the body of the sauce, and some that retain more texture.
For example, Cook's Illustrated recommends Jonagold, Jonathan, Pink Lady, and Macoun (possible pay wall).
In an article at Oregon Live, interviewing applesauce expert Peggy Acott, writes:
"Even two sweet apples taste different," [Peggy Acott] says. "You can also pair
a sweet and a tart apple, a mild and a spicy one, or spicy, sweet and
mild ones." She suggests using two or three varieties. "One year I
used four and that was overkill."
She believes in tasting the apples before you choose them. The same
variety of apple will taste different year to year, she says. Some of
her favorite past pairings include Ginger Goldens, a creamy yellow
sweet-tart apple, and at least one other variety. Maybe Cox's Orange
Pippins, an old English variety known for being tart and crisp, or a
Rubinette (a cross of Golden Delicious and Cox's Orange Pippin).
An article at the Your pick of apples: Which varieties work best for baking, sauces and other recipes at the Missoulian indicates:
The spicy, supple McIntosh will melt like ice cream when baked, but
creates a smooth, flavorful applesauce. The soft, tangy Jonathan and
the sweet, crisp Empire will also deliver a flavorful puree. The Cox’s
Orange Pippin [...] is a wonderful juicy heirloom for sauce.
As you can see, in the end, you will want to get to know your own locally available apple varieties, and what you like. There is a tremendous amount of variation.
Apple sauce is very simple and quick to make, especially in the microwave. If you are a huge applesauce fan, you might want to make a sample from one apple of each variety and take notes on its flavor and texture.
Best Answer
It's not going to hurt anything. The apples they use for cider are usually pretty rough, so a little wrinkling isn't going cause a health issue. You very well may not get as much juice, but the juice you will get will be more concentrated.
The same principle applies to grapes used for wine...Ideally they will get very little water in the weeks leading up to harvest, so you'll get all the good sugars and flavours, and less of the undesirable water.